Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$

HINT:

For $(1),$ Using If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.,

we have $x^3+y^3+z^3=3xyz$

Now $x^3=e^{i(3a)}=\cos3a+i\sin3a$

and $xy=e^{i(a+b)}=\cos(a+b)+i\sin(a+b),xyz=\cdots$

For $(3),(4)$

$x+y+z=0\implies\cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0$

So, $x^{-1}+y^{-1}+z^{-1}=?$

Now $x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+zx)=(x+y+z)^2-2xyz(x^{-1}+y^{-1}+z^{-1})=?$

A hint:

You have three unit vectors in the plane, summing up to $0$. How does the figure look like when you add them geometrically as you add forces?

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